Control of industrial robots. Control of the interaction

Transcription

1 Control o industrial robots Control o the interation Pro. Paolo Roo (paolo.roo@polimi.it) Politenio di Milano Dipartimento di Elettronia, Inormazione e Bioingegneria

2 Introdution So ar we have assumed that the robot is ontrolled in a ree environment However the robot has to interat with the environment either to manipulate it, or to avoid to ollide with it, or to interat with other devies or humans In order to eetively interat with the environment, the robot must be endowed with eternal (eteroeptive) sensors, whih make the robot ontroller aware o the situation Common sensors used to measure suh interation are ore sensors and vision sensors In the ollowing we will study the ontrol systems based on ore measurements. Control o industrial robots Control o the interation Paolo Roo

3 Interation with the environment A robot ommonly interats with the working environment, manipulating objets and perorming operations on suraes (think o grinding, deburring, part assembly). In modern robotis, situations are ommon where the robot physially interats with the operator (think o telemanipulation or rehabilitation). Control o industrial robots Control o the interation Paolo Roo

4 Interation with the environment We are now dealing with the modelling o the environment and the use o eternal sensors: eternal sensors environment task planner trajetory planner ontrol robot internal sensors Control o industrial robots Control o the interation Paolo Roo

5 Passive ontrol o ompliane A irst way to address the interation with the environment is to endow the manipulator with devies that ailitate the eeution o the task in a passive way: RCC Remote Center o Compliane The RCC is used in assembly tasks (peg-in-a-hole) It is typially plaed between the robot s wrist and the gripper The RCC lets the gripper assembly move in the plane perpendiular to the peg s ais. This allows the peg to rotate With the RCC, the ores generated by any misalignment an be ompensated in a passive way Control o industrial robots Control o the interation Paolo Roo

6 Ative ontrol o the interation During more general interation tasks, the use o a purely positional ontrol strategy (the same adopted in ree motion) may lead to problems due to positioning errors and unertainties in task planning, related to a inomplete knowledge o the environment. In many appliations ontrolling the ontat ore is as important as ontrolling the speed o the robot alongthe presribed diretiont. Control o industrial robots Control o the interation Paolo Roo

7 Ative ontrol o the interation Eamples where an ative ontrol o the interation is desirable inlude: mehanial mahining (deburring, surae inishing, polishing, assembly,...) tele-manipulation ooperation o multi-manipulator systems deterous robot hands physial human-robot interation For an ative ontrol o interation it is neessary to use devies or ore (and moment) measurement. Control o industrial robots Control o the interation Paolo Roo

8 Fore sensors Fore/torque sensors are devies that return the measurements o three omponents o ore and three omponents o moment with respet to a loal rame. Control o industrial robots Control o the interation Paolo Roo

9 Internal behaviour Sensors are based on strain gauges, i.e. devies that an measure a strain. The strain indues a variation in the resistane o a iruit. Strain gauges are suitably mounted, in suh a way as to return inormation rom whih three omponents o ore and moment an be omputed. Control o industrial robots Control o the interation Paolo Roo

10 Statis o the manipulator When studying the interation o the robot with the environment, the problem arises how to inlude the eet o a ore applied at the end-eetor in the model. Suppose then that a ore is applied at the endeetor. What are the joint torques that keep the system in equilibrium???? Control o industrial robots Control o the interation Paolo Roo

11 Systems o ores As a matter o at a system o ores may at at the end-eetor. How an this system be redued? From lassial mehanis One ore applied in a point (the resultant) One torque It ollows that we will always be able to represent a system o ores with two vetors: A ore vetor applied in a point O (the resultant) A moment vetor (the moment o the original system o ores with respet to the ulrum O) Control o industrial robots Control o the interation Paolo Roo

12 Statis o the manipulator Let us then onsider a manipulator subjet to a set o ores at the end eetor, having a resultant and moment, w.r.t. the origin o the end eetor rame, µ. We then deine vetor: h = µ The virtual work o the ores/torques at the joints is: T δw τ = τ δq T T δw = δp + µ ωdt = J µ h Let us all τ the vetor o torques (or ores) applied to the joints. where δq is the elementary displaement o the joint oordinates. The virtual work o the ores at the end-eetor is: T T ( q) δq + J ( q) δq = h J( q) δq Control o industrial robots Control o the interation Paolo Roo T P where δp is the virtual displaement o the origin o the end-eetor rame, onsistent with the dierential kinematis o the robot, and ω is the angular veloity o the end eetor rame. O

13 Statis o the manipulator T δw τ = τ δq T δw = J µ h P T T ( q) δq + J ( q) δq = h J( q) δq, O J ( q) = J J P O ( q) ( ) q position Jaobian orientation Jaobian Based on the priniple o virtual work the manipulator is in stati equilibrium i and only i: δw τ = δw h τ = J T ( q)h Notie that the same matri, the Jaobian matri, whih deines the dierential kinematis o the manipulator, is also involved in the statis o the system (kinetostati duality). Control o industrial robots Control o the interation Paolo Roo

14 Taonomy Fore based interation ontrol strategies an be divided into two ategories: Impedane/admittane ontrol The goal is to assign a presribed dynami relation between interation ores and position errors We want that the manipulator, in ase o unplanned interations, behaves like a generalized mass-spring-damper system Hybrid position and ore ontrol We separate diretions whih are only position ontrolled rom diretions whih are only ore/torque ontrolled In the diretions onstrained by the environment we want that a ore/torque with a speiied value be established Furthermore we will talk about: Epliit ontrol The ore (or the impedane) is diretly assigned, ating on the ontrol variables (joint torques) Impliit ontrol The ore (or the impedane) is indiretly assigned, ating on the set-points o the position ontrol loops Control o industrial robots Control o the interation Paolo Roo

15 Mehanial impedane The low o power among two physial systems an always be deined as the produt o two onjugate quantities: a (generalized) eort and a (generalized) low In the eletrial ramework, the low is the urrent, while the eort is the voltage. In the mehanial ramework, instead, the low is the (linear or angular) veloity, the eort is the ore (or the moment) The mehanial impedane is then deined as dynamial relation that is established between ore and veloity (or displaement) or a mehanial system. The admittane is just the reiproal o the impedane The impedane ontrol aims at making the manipulator, position ontrolled and in interation with the environment, assume a desired mehanial impedane, like a generalized mass-spring-damper system Control o industrial robots Control o the interation Paolo Roo

16 Mehanial impedane Eamples o impedane/admittane ontrol or physial human-robot interation: Control o industrial robots Control o the interation Paolo Roo

17 A 1 d.o.. system In order to understand how an impedane ontroller operates, let us irst onsider the simple eample o a body with mass M interating with the environment, whih eerts a ore : p u M The model o the system is simply: Ma = u + Notie that the model o the interation with the environment is not speiied: we just know that there is an eternal ore ating on the system Control o industrial robots Control o the interation Paolo Roo

18 A 1 d.o.. system I we assume a ontrol proportional to position and veloity: u = k1p k2v We obtain: Ma k v + k p = k 1 k 2 p M The ontrolled system reats to the eternal ore like a mass-spring-damper The spring and damper ators are tunable, the mass is the physial one o the system. Control o industrial robots Control o the interation Paolo Roo

19 A 1 d.o.. system I we want to hange the mass, too, we need to measure the ore: u M M ( k p k v + ) = 1 2 d We obtain: M d k 1 M d a k v + k p = k 2 p All the parameters o the mass, spring and damper system are now tunable. Control o industrial robots Control o the interation Paolo Roo

20 Etension to the whole manipulator We now want to etend the impedane ontrol to the whole manipulator. The dynami model in interation with the environment is given by the equation: ( ) q T + C( q, q ) q + g( q) = τ J ( q)h B q where h is the vetor o the ontat ores (and moments) applied by the end eetor on the environment. Let us onsider now the adoption o an inverse dynamis ontrol law: τ ( q) y + C( q, q ) q g( q) = B + Substituting we obtain: nonlinear oupling term q = y B 1 T ( q) J ( q)h Control o industrial robots Control o the interation Paolo Roo

21 Inverse dynamis ontrol Assume now or y the epression introdued in the inverse dynami ontrol in the operational spae: y By substituting the epression or y we obtain the equation o the dynamis in losed loop: ~ where = d, while: B A 1 = J A ( ) q q ( q) + K ~ + K ~ J ( q, ) d T 1 ( q) J ( q) B( q) J ( q) D ~ K P ~ 1 K D ~ + + = B A = A A P ( q) h A is the inertia matri in the operational spae and: A generalized ores perorming work on ẋ h A = T T A ( ) h, J = T A ( ) J A T J h = J T A h A Control o industrial robots Control o the interation Paolo Roo

22 Impedane ontrol with ore measurement ~ K P ~ 1 K D ~ + + = B A ( q) h A We have obtained an impedane relation whih is oupled and only partially assignable: a ore applied along a Cartesian diretion yields motion in other diretions as well. I ore measurements are available, we an set: τ T ( ) y + C( q, q ) q + g( q) + J ( q)h = B q y = J 1 A ( ) q q h 1 ( q) M M + D ~ + K ~ M J ( q, ) d d d d d d A A where M d, D d, K d are diagonal, positive deinite matries. We obtain: M ~ + D ~ + K ~ = d d d h A i.e. a ompletely deoupled system. Control o industrial robots Control o the interation Paolo Roo

23 Impedane ontrol The equation: M ~ + D ~ + K ~ = d d d h A deines a mehanial impedane relation between the ores/moments deined at the right hand side o the equations and the position/orientation errors in the operational spae. This impedane has the orm o a mehanial system with mass matri M d, damping D d, and stiness K d. The dynami behaviour along eah diretion o the operational spae an thus be speiied: a ore or a moment along a ertain Cartesian diretion yields a motion in suh diretion, whih is related to the ore and moment by a ully ontrollable dynamis. But what do we eatly mean or impedane in the operational spae? Control o industrial robots Control o the interation Paolo Roo

24 Translational impedane The irst three degrees o reedom in the operational spae are the position oordinate o the end-eetor. In this ase the deinition o the impedane is rather intuitive: p d : desired position p e : atual position p = p d p e The impedane relation is: M p + D p + K p = p p p We thus apply to eah degree o reedom what we have already seen or the single mass. Control o industrial robots Control o the interation Paolo Roo

25 Rotational impedane The remaining three degrees o reedom in the operational spae onsist in a minimal representation (Euler angles) o the end-eetor orientation. In this ase the deinition o the impedane is less intuitive: φ d : desired orientation φ e : atual orientation φ = φ d φ e The impedane relation is: M φ + D φ + K φ = φ φ φ Τ T ( φ )µ e Due to the presene o matri T, this relation is subjet to representation singularities. Better deinition o the rotational impedane have been proposed in the literature. Control o industrial robots Control o the interation Paolo Roo

26 Impliit impedane ontrol (admittane ontrol) When a positional ontrol is already applied to the mehanial system, it might be onvenient to lose the impedane ontrol outside the position ontrol. Consider again the ase o single d.o.. system, position ontrolled: d R u M The system an be modeled with the blok diagram: u d R(s) G(s) 1 ( s) 2 G = Ms Control o industrial robots Control o the interation Paolo Roo

27 Impliit impedane ontrol (admittane ontrol) Consider the ollowing blok diagram, where the ore applied by the environment is measured and iltered through a system with transer untion G d (s), to give a variation o the reerene d : d + + R(s) u + + G(s) G d (s) impedane ontroller Elaborating the blok diagram: = G s 1+ R s ( ) ( ) G( s) + ( G ( s) ) d d ( ) G( s) ( ) G( s) R s 1+ R s Also alled admittane ontroller (it aepts a ore and delivers a displaement) Control o industrial robots Control o the interation Paolo Roo

28 Impliit impedane ontrol (admittane ontrol) In a band o requenies well inside the bandwidth o the position ontrol system, we an assume the ore disturbane well rejeted and the position reerene orretly traked. Thus: d Gd ( s) = G ( s) 1~ d I we impose: G d ( s) = M d s D d s + K d M ~ d + Dd ~ + Kd ~ = we assign to the manipulator the desired impedane behaviour. The advantage o this sheme is that the position ontrol is kept unhanged: the impedane ontrol modiies the set-points o the position ontrol, whih is usually the only openness allowed by an industrial motion ontrol system. Control o industrial robots Control o the interation Paolo Roo

29 Admittane ontrol or the whole manipulator The same sheme as with the single mass an be used, applied to eah Cartesian diretion o the end-eetor: p p d p d + R u, y, z The approah an be generalized to aount also or moments ating at the end-eetor and onsequent variations to the desired orientation o the end-eetor F Admittane ilter Control o industrial robots Control o the interation Paolo Roo

30 Admittane ontrol used or manual guidane Control o industrial robots Control o the interation Paolo Roo

31 Natural onstraints and artiiial onstraints An alternative way to perorm an interation task is to assign reerene values o the ores and o the positions, onsistent with the geometry o the environment. Some diretions an be in at subjeted to onstraints on the position, other ones to onstraints on the ores that an be applied. In general, it is not possible to impose both the ore, and the position, along the same diretion. We make the ollowing distintion: Natural onstraints: Artiiial onstraints: they are imposed by the environment along eah degree o reedom o the task; they depend on the geometry o the task. onstraints imposed by the ontrol system, related to the task eeution strategy. For task desription we reer to a onstraint rame, y, z. Control o industrial robots Control o the interation Paolo Roo

32 Sliding on a lat surae Natural Constraints p ω ω z y y µ z Artiiial Constraints µ z µ p p ω y y z This piture is taken rom the tetbook: B. Siiliano, L. Siavio, L. Villani, G. Oriolo: Robotis: Modelling, Planning and Control, 3rd Ed. Springer, 2009 Control o industrial robots Control o the interation Paolo Roo

33 Insertion o a peg in a hole This piture is taken rom the tetbook: B. Siiliano, L. Siavio, L. Villani, G. Oriolo: Robotis: Modelling, Planning and Control, 3rd Ed. Springer, 2009 Natural Constraints p p ω ω y y z µ z Artiiial Constraints. Usually we set p z 0 and all the other artiiial onstraints to zero. µ µ y p ω y z z Control o industrial robots Control o the interation Paolo Roo

34 Rotation o a rank Oriented like the rank Natural Constraints p p ω z Artiiial Constraints z µ Idle knob ω y y µ z µ p ω y y z This piture is taken rom the tetbook: B. Siiliano, L. Siavio, L. Villani, G. Oriolo: Robotis: Modelling, Planning and Control, 3rd Ed. Springer, 2009 Normally artiiial onstraints on ores and moments are assumed to be zero. Control o industrial robots Control o the interation Paolo Roo

35 Seletion matri We assume that the onstraints an always be epressed along mutually orthogonal diretions o rame C. Thus on eah diretion either a natural onstraint on position, or a natural onstraint on ore, eists. In order to speiy the onstraints we use a seletion matri Σ diagonal and with null or unitary elements: Natural onstraints Artiiial onstraints n ( I ) h h n Σv = v Σ = ( I Σ) v = v Σh = h a Σ = a The seletion matri etrats diretions where a natural onstraint in veloity applies and thus an artiiial onstraint in ore an be enored Control o industrial robots Control o the interation Paolo Roo

36 Hybrid ore/position ontrol Hybrid ore/position ontrol is a ontrol strategy that uses the artiiial onstraints to speiy the goals o the ontrol system, in terms o both ore and position. The 6 diretions o the operational spae are divided into position/veloity ontrolled diretions and ore/moment ontrolled diretions: Σ h Ad Σ INTERACTION CONTROL y F + y NONLINEAR u MANIPULATOR h A v d I Σ MOTION CONTROL y v + COMPENSATIONS AND ENVIRONMENT v All quantities reerred to the rame C I Σ Control o industrial robots Control o the interation Paolo Roo

37 Soures o inonsisteny Hybrid position/ore ontrol is based on a nominal model o the interation. Inonsisteny may however our in the measurements, due e.g. to: 1. rition at the ontat (a ore is deteted in a nominally ree diretion) 2. ompliane in the robot struture and/or at the ontat (a displaement is deteted in a diretion whih is nominally onstrained in motion) 3. unertainty in the environment geometry at the ontat The irst two soures o inonsisteny are automatially iltered out through the seletion matries. The third soure o inonsisteny an be mitigated by real time estimation proess. Control o industrial robots Control o the interation Paolo Roo

38 A 1 d.o.. system In order to have a better idea on how the ontrol system operates along the ore ontrolled diretions, let s onsider the simple eample o a body o mass M in interation with the environment: u M We assume a purely elasti model o the environment. Blok diagram: u + G(s) K u G (s) G K ( s) = 2 Ms + K Control o industrial robots Control o the interation Paolo Roo

39 A 1 d.o.. system u G (s) G 2 K n ( s) = ω = ωn = Ms + K s + ω, n K M The system presents a unitary gain and resonant, high requeny (or a sti ontat) dynamis. Notie that = u or K. The best way to ontrol this system is through an integral ontroller: d + u R R (s) G (s) ( s) = K s i ω K i This way the bandwidth o the ore ontrol is diretly assigned with the integrator, leaving outside the bandwidth the high requeny dynamis o the interation with the environment. Control o industrial robots Control o the interation Paolo Roo

40 Impliit ore ontrol As with impedane ontrol, when a position ontrol is already ative, it might be onvenient to lose the ore ontrol around the position ontrol loop. Consider or simpliity only the ase o a, position ontrolled, single degree o reedom: d R u M The system an be modelled through the ollowing blok diagram: d + R(s) u + G(s) K The advantage is that the position ontrol is let unhanged. Control o industrial robots Control o the interation Paolo Roo

41 Impliit ore ontrol Solving the blok diagram: ( s) ( s) = d 1+ G s + R( s) G( s) K ( ) K R( s) G( s) Under the assumption o an ininitely sti ontat (K ) we obtain =0 and u: d R(s) u 1 The position ontrol loop gets open! d I a PID position ontroller is used: ( ) R s we an adopt or the ore ontrol the sheme: + R (s) C(s) d 2 KD s + KPs + K = s u R(s) 1 I ( ) C s = K Partially ompensates the eet o the PID ontroller D s K P s + K I Control o industrial robots Control o the interation Paolo Roo

42 Impliit ore ontrol Then, ater ompensating the eet o the PID ontroller: d + R (s) 1 s We an take R as a PI ontroller: ( s) R = Kp + K s i Tuning o this ontroller is straightorward: ω K p L sk p ( s) = 2 s + K i K i K p Control o industrial robots Control o the interation Paolo Roo

43 Industrial appliation o ore ontrol ABB RobotWare Mahining Fore Control (mahining o blades): Control o industrial robots Control o the interation Paolo Roo